1998 Mercury Marquis Battery light comes on intermittedly

Electrical problem
1998 Mercury Marquis 6 cyl Two Wheel Drive 103000 miles

My battery light came on and I took it to be serviced and the alternator was replaced and the battery was good. However after 250 miles of high way driving, the battery light came on again. The light went off the next day. Took it be serviced and all connections were checked. No problem.
However after about 10 miles, the battery came on againg.
After sitting for a whole day, the car started with no problem but after 1 mile, the light came on and then after 15 miles, it went off.